If 3(log 5 - log 3) - (log 5 - 2 log 6) = 2 - log x, find x.

Answer

To solve the equation \( 3(\log 5 - \log 3) - (\log 5 - 2 \log 6) = 2 - \log x \), we will follow these steps:

### Step 1: Expand the left side of the equation
Start by distributing the terms in the left-hand side:

\[ 3(\log 5 - \log 3) - (\log 5 - 2 \log 6) = 3 \log 5 - 3 \log 3 - \log 5 + 2 \log 6 \]

### Step 2: Combine like terms
Now, combine the logarithmic terms:

\[ (3 \log 5 - \log 5) + 2 \log 6 - 3 \log 3 = 2 \log 5 + 2 \log 6 - 3 \log 3 \]

### Step 3: Rewrite \( \log 6 \)
We can express \( \log 6 \) in terms of \( \log 2 \) and \( \log 3 \):

\[ \log 6 = \log(2 \cdot 3) = \log 2 + \log 3 \]

Thus, \( 2 \log 6 = 2(\log 2 + \log 3) = 2 \log 2 + 2 \log 3 \).

### Step 4: Substitute back into the equation
Substituting this back into our equation gives:

\[ 2 \log 5 + 2 \log 2 + 2 \log 3 - 3 \log 3 = 2 \log 5 + 2 \log 2 - \log 3 \]

### Step 5: Set the equation equal to the right side
Now we have:

\[ 2 \log 5 + 2 \log 2 - \log 3 = 2 - \log x \]

### Step 6: Move \( \log x \) to the left side
Rearranging gives:

\[ 2 \log 5 + 2 \log 2 - \log 3 + \log x = 2 \]

### Step 7: Combine logarithms
Using the property \( \log a + \log b = \log(ab) \):

\[ \log(5^2) + \log(2^2) + \log x - \log 3 = 2 \]

This simplifies to:

\[ \log\left(\frac{25 \cdot 4 \cdot x}{3}\right) = 2 \]

### Step 8: Exponentiate to eliminate the logarithm
Taking the antilogarithm gives:

\[ \frac{25 \cdot 4 \cdot x}{3} = 10^2 \]

This simplifies to:

\[ \frac{100x}{3} = 100 \]

### Step 9: Solve for \( x \)
Multiplying both sides by 3:

\[ 100x = 300 \]

Now, divide by 100:

\[ x = 3 \]

Thus, the value of \( x \) is \( \boxed{3} \).

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